Integral type Fixed point Theorems for α-admissible Mappings Satisfying α-ψ-φ-contractive Inequality

Transcription

1 Filomat 3:12 (216, DOI /FIL B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Integral type Fixed point Theorems for α-admissible Mappings Satisfying α-ψ-φ-contractive Inequality Ziad Badehian a, Mohammad Sadegh Asgari a a Department of Mathematics, IslamicAzad University, Central Tehran Branch, Iran Abstract. In this paper, we establishe some new fixed point theorems by α-admissible mappings satisfying α-ψ-φ-contractive inequality of integral in complete metric spaces. Presented results can be considered as an extension of the theorems of Banach-Cacciopoli and Branciari. 1. Introduction and Preliminaries The first well known result on fixed points for contractive mapping was Banach-Cacciopoli theorem(1922[1], as follow: Theorem 1.1. Let (X, d be a complete metric space, c (, 1, and let f : X X be a mapping such that for each x, y X, d( f x, f y cd(x, y (1 Then, f has a unique fixed point a X such that for each x X, lim n f n x = a. Branciari(22[2], proved the following theorem: Theorem 1.2. Let (X, d be a complete metric space, c (, 1, and let f : X X be a mapping such that for each x, y X, d( f x, f y d(x,y φ(tdt c φ(tdt (2 where φ : [, [, is a Lebesgue-integrable map which is summable, (i.e., with finite integral on each compact subset of [,, nonnegative, and such that for each ɛ >, φ(tdt > ; then f has a unique fixed point a X such that for each x X, lim n f n x = a. Samet et al. (212[7] have introduced α-ψ-contractive type mappings and also α-admissible functions in complete metric spaces. In this paper, we introduce α-admissible mappings satisfying α-ψ-φ-contractive integral type inequality and we establish some fixed point theorems in complete metric spaces. Our results can be considered as an extension of Banach-Cacciopoli and Branciari theorems. 21 Mathematics Subject Classification. Primary 47H1 ; Secondary 26D15 Keywords. Fixed point theorem, α-admissible mapping, α-ψ-φ-contractive mapping of integral type, complete metric space. Received: 17 June 214; Accepted: 2 April 215 Communicated by Dragan S. Djordjević addresses: Ziadbadehian@gmail.com (Ziad Badehian, moh.asgari@iauctb.ac.ir (Mohammad Sadegh Asgari

2 Z. Badehian, M. S. Asgari / Filomat 3:12 (216, Definition 1.3. [7] Let α : X X [, be a function, we say f : X X, is α-admissible if for all x, y X, α(x, y 1 = α( f (x, f (y 1 (3 Definition 1.4. [7] Let Ψ be a family of nondecreasing functions ψ : [, [, such that for each ψ Ψ and t >, n=1 ψn (t < +. where ψ n is the n-th iterate of ψ. Lemma 1.5. ([7] If ψ : [, [, is nondecreasing function and for each t >, lim n ψ n (t = then ψ(t < t. Definition 1.6. Let Φ be a collection of mappings φ : [, [, which are Lebesgue-integrable, summable on each compact subset of [, and satisfying following condition: φ(tdt >, f or each ɛ >. Lemma 1.7. ([3] Let φ Φ and {r n } n N be a nonnegative sequence with lim n r n = a, then rn lim n φ(tdt = a φ(tdt. Lemma 1.8. ([3] Let φ Φ and {r n } n N be a nonnegative sequence. Then rn lim n φ(tdt = lim n r n =. 2. Fixed Point Theorems Definition 2.1. Let (X, d be a metric space and f : X X be a given mapping. We say that f is an α-ψ-φcontractive integral type mapping if there exist three functions α : X X [, +, φ Φ and ψ Ψ such that α(x,yd( f x, f y φ(tdt ψ for all x, y X and all t [, +. ( d(x,y φ(tdt, (4 Theorem 2.2. Let (X, d be a complete metric space and f : X X be a mapping satisfying following conditions: (i f is α-admissible; (ii there exists x X such that α(x, f x 1; (iii f is an α-ψ-φ-contractive integral type mapping, then f has a unique fixed point a X such that for x X, lim n f n x = a. Proof. First, we show that d( f n x, f n+1 x ( d(x, f x φ(tdt ψ n φ(tdt. (5 Since f is α-admissible, from (ii we get α(x, f x 1 α( f x, f 2 x 1... α( f n x, f n+1 x 1, then for all n N, α( f n x, f n+1 x 1. (6

3 Z. Badehian, M. S. Asgari / Filomat 3:12 (216, To prove (5, by induction and (i, we have for n = 1 d( f x, f 2 x φ(tdt α(x, f x d( f x, f 2 x φ(tdt ( d(x, f x ψ φ(tdt. Now, assuming that (5 holds for an n N, then from (6 we get d( f n+1 x, f n+2 x φ(tdt α( f n x, f n+1 x d( f n+1 x, f n+2 x d( f n x, f n+1 x ψ φ(tdt ( d(x, f x ψ (ψ n φ(tdt ( d(x, f x = ψ n+1 φ(tdt, φ(tdt therefore (5 holds for all n N. Now, if n + we obtain d( f n x, f n+1 x φ(tdt, then, from Lemma 1.8 we have d( f n x, f n+1 x. In the following, we show that { f n x } is a Cauchy sequence. Suppose that there exists an ɛ > such that n=1 for each k N there are m k, n k N with m k > n k > k, such that Hence d( f m k x, f n k x ɛ, d( f m k 1 x, f n k x < ɛ. (7 ɛ d( f m k x, f n k x d( f m k x, f m k 1 x + d( f m k 1 x, f n k x Letting k + then we have < d( f m k x, f m k 1 x + ɛ. d( f m k x, f n k x ɛ +, (8 this implies that there exists l N, such that k > l = d( f ml+1 x, f nl+1 x < ɛ. (9 Actually, if there exists a subsequence {k l } N, k l > l, d( f m k l +1 x, f n k l +1 x ɛ, if l + we get ɛ d( f m k l +1 x, f n k l +1 x d( f m k l +1 x, f m k l x + d( f m k l x, f n k l x + d( f n k l x, f n k l +1 x < ɛ

4 therefore, from (4 and lemma 1.5 we have d( f m kl +1 x, f n k l +1 x Z. Badehian, M. S. Asgari / Filomat 3:12 (216, φ(tdt ψ letting l +, then we obtain φ(tdt < φ(tdt, < d( f m kl x, f n k l x m d( f kl x, f n k l x φ(tdt φ(tdt, which is a contradiction. Then, (9 holds. Now, we prove that there exists σ ɛ (, ɛ, k ɛ N such that k > k ɛ = d( f m k+1 x, f n k+1 x < ɛ σ ɛ. (1 Assume that (1 is not true, from (9 there exists a subsequence {k l } N, d( f m k l +1 x, f n k l +1 x ɛ, as l +. Then from (4 and lemma 1.5, we get d( f m kl +1 x, f n k l +1 x φ(tdt ψ Suppose that l + then we obtain φ(tdt < φ(tdt, d( f m kl x, f n k l x d( f φ(tdt < m kl x, f n k l x φ(tdt. which is a contradiction. So, (1 holds. Therefore, { f n x } n=1 is a Cauchy sequence. In fact, for each k > k ɛ we have ɛ d( f m k x, f n k x d( f m k x, f m k+1 x + d( f m k+1 x, f n k+1 x + d( f n k+1 x, f n k x < d( f m k x, f m k+1 x + (ɛ σ ɛ + d( f n k+1 x, f n k x ɛ σ ɛ as k +. which is a contradiction. Since (X, d is a complete metric space, then there exists a X such that lim n f n x = a. Now, we show that a is a fixed point of f. We claim that d( f a, a d( f a, f ( f n x + d( f n+1 x, a to prove our claim it is sufficient to show that, d( f a, f ( f n x. We have < d( f a, f ( f n x φ(tdt if n + then d( f a, f ( f n x φ(tdt, α(a, f n x d( f a, f ( f n x ( d(a, f n x ψ φ(tdt < d(a, f n x φ(tdt, φ(tdt

5 consequently, from lemma 1.8 we obtain d( f a, f ( f n x. Z. Badehian, M. S. Asgari / Filomat 3:12 (216, Therefore, d( f a, a = and so f a = a, which means a is a fixed point of f. a is a unique fixed point. Let b another fixed point of f, then we have d(a,b φ(tdt = d( f a, f b φ(tdt α(a,bd( f a, f b ψ < ( d(a,b d(a,b φ(tdt φ(tdt φ(tdt which is impassible. Then, d(a, b = and so a = b. In the following, we define subclass of integrals and we prove the existence of fixed point by applying these integrals. Definition 2.3. Let Γ be a collection of mappings γ : [, [, which are Lebesgue-integrable, summable on each compact subset of [, and satisfying following conditions: (1 γ(tdt >, for each ɛ > (2 a+b γ(tdt a γ(tdt + b γ(tdt, for each a, b Definition 2.4. Let (X, d be a metric space and f : X X be a given mapping. We say that f is an α-ψ-γ-contractive integral type mapping if there exist three functions α : X X [, +, γ Γ and ψ Ψ such that α(x,yd( f x, f y γ(tdt ψ for all x, y X and all t [, +. ( d(x,y γ(tdt, (11 Theorem 2.5. Let (X, d be a complete metric space and f : X X be a mapping satisfying following conditions: (i f is α-admissible; (ii there exists x X such that α(x, f x 1; (iii f is an α-ψ-γ-contractive integral type mapping, then f has a unique fixed point a X such that for x X, lim n f n x = a. Proof. First, we show that d( f n x, f n+1 x ( d(x, f x γ(tdt ψ n γ(tdt. (12 Since f is α-admissible then from (ii we get α(x, f x 1 α( f x, f 2 x 1... α( f n x, f n+1 x 1. then for all n N, α( f n x, f n+1 x 1. (13

6 Z. Badehian, M. S. Asgari / Filomat 3:12 (216, To prove (12,by induction and (i, we have for n = 1 d( f x, f 2 x γ(tdt α(x, f x d( f x, f 2 x γ(tdt ( d(x, f x ψ γ(tdt. Now, assuming that (12 holds for an n N, then from (13 we get d( f n+1 x, f n+2 x γ(tdt ψ α( f n x, f n+1 x d( f n+1 x, f n+2 x d( f n x, f n+1 x ψ γ(tdt ( d(x, f x ψ (ψ n γ(tdt ( d(x, f x = ψ n+1 γ(tdt, γ(tdt therefore (12 holds for all n N. In the following, we show that { f n x } is a Cauchy sequence. Fix ɛ > n=1 and let n(ɛ N such that ( d(x, f x ψ n γ(tdt < ɛ. n n(ɛ Let m, n N with m > n > n(ɛ, we get d( f n x, f m x which means d( f n x, f m x γ(tdt d( f n x, f n+1 x + +d( f m 1 x, f m x d( f n x, f n+1 x γ(tdt d( f m 1 x, f m x γ(tdt + + γ(tdt ( d(x, f x ( d(x, f x ψ n γ(tdt + + ψ m 1 γ(tdt m 1 ( d(x, f x = ψ k γ(tdt k=n n n(ɛ γ(tdt, therefore, by lemma 1.8 we have d( f n x, f m x. ( d(x, f x ψ n γ(tdt < ɛ, Then { f n x } is a Cauchy sequence. Since (X, d is a complete metric space, then there exists a X such n=1 that, lim n f n x = a. Now, we show that a is a fixed point of f. We claim that d( f a, a d( f a, f ( f n x + d( f n+1 x, a.

7 Z. Badehian, M. S. Asgari / Filomat 3:12 (216, To prove our claim, it is sufficient to show that d( f a, f ( f n x. We have < d( f a, f ( f n x if n + then d( f a, f ( f n x γ(tdt α(a, f n x d( f a, f ( f n x ( d(a, f n x ψ γ(tdt γ(tdt, d(a, f n x consequently, from lemma 1.8 we obtain d( f a, f ( f n x. γ(tdt, γ(tdt Therefore, d( f a, a = and so f a = a, which means a is a fixed point of f. The proof of the uniqueness of fixed point is similar to theorem Examples and Remarks Example 3.1. Let X = R and d(x, y = x y for all x, y R, then (X, d is a complete metric space. Define f : R R by f x = x 2 and α : R R [, + by 4 3 if x, y [, 1]; α(x, y = o.w. f is α-admissible, since α(x, y 1 implies that ( x α( f x, f y = α 2, y 1. 2 Define φ : [, [, by φ(t = t then φ Φ. There exists x R such that α(x, f x = α(x, x 2 1. Let ψ(t = t 2 where ψ : [, [, then, for all x, y R we have α(x,yd( f x, f y φ(tdt = = ψ 4 3 d( x 2, y 2 d(x,y 2 tdt ( d(x,y tdt φ(tdt. Then all the hypotheses of Theorem 2.2 are satisfied, consequently f has a unique fixed point. Here, is a fixed point of f (x. Example 3.2. Let X = R + and d(x, y = x y for all x, y R +, then (X, d is a complete metric space. Define f : R + R + by f x = x and α : R+ R + [, + by 5 4 if x, y [, 2]; α(x, y = o.w

8 f is α-admissible, since α(x, y 1 implies that ( x α( f x, f y = α 3 + 1, y Z. Badehian, M. S. Asgari / Filomat 3:12 (216, Define γ : [, [, by γ(t = 2 then γ Γ. There exists x X such that α(x, f x = α(x, x 3 instance let x =. Let γ(t = t 2 where γ : [, [, then for all x, y X we have α(x,yd( f x, f y γ(tdt = 5 4 d( x 3 +1, y 3 +1 = 5 d(x, y 6 d(x,y 2dt 2 = d(x, y = ψ ( d(x,y 2dt γ(tdt , for Then all the hypotheses of Theorem 2.8 are satisfied, consequently f has a unique fixed point. Here, 3 2 is a fixed point of f (x. Remark 3.3. In theorem 2.2, if we consider f : X X which is α-admissible where α(x, y = 1 for all x, y X and ψ(t = ct for all t > and some c (, 1 then we get Branciari theorem. Remark 3.4. Moreover, by above conditions if φ(t = 1 or γ(t = 1 in theorem 2.5 then we conclude Banach- Caccioppoli principle. In fact, we have d( f x, f y d(a,b 1dt = d( f x, f y cd(x, y = c 1dt. Acknowledgments We are grateful to the referees for their valuable comments that improved the quality of the paper. References [1] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fund. Math 3 ( [2] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci 29 ( [3] Z. Liu, X. Li, S. M. Kang, S. Y. Cho, Fixed point theorems for mappings satisfying contractive conditions of integral type and applications, Fixed Point Theory Appl No. 64 ( pages doi: / [4] Mocanu, M, Popa, V: Some fixed point theorems for mappings satisfying implicit relations in symmetric spaces, Libertas Math 28 ( [5] V. L. Rosa, P. Vetro, Common fixed points for α-ψ-φ-contractions in generalized metric spaces, Nonlinear Analysis: Modelling and Control 19 No. 1 ( [6] B. Samet, A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type, Int. J. Math. Anal 3 ( [7] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal 75 ( [8] M. Samreen, T. Kamran, Fixed point theorems for integral G-contractions, Fixed Point Theory and Applications (